Groups acting on quasi-median graphs. An introduction
Anthony Genevois1
1 Université Paris-Sud, Orsay, (France)
Séminaire de théorie spectrale et géométrie, Volume 35 (2017-2019), pp. 43-68.

Quasi-median graphs have been introduced by Mulder in 1980 as a generalisation of median graphs, known in geometric group theory to naturally coincide with the class of CAT(0) cube complexes. In his PhD thesis, the author showed that quasi-median graphs may be useful to study groups as well. In the present paper, we propose a gentle introduction to the theory of groups acting on quasi-median graphs.

Published online:
DOI: 10.5802/tsg.363
Anthony Genevois 1

1 Université Paris-Sud, Orsay, (France) 
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Anthony Genevois. Groups acting on quasi-median graphs. An introduction. Séminaire de théorie spectrale et géométrie, Volume 35 (2017-2019), pp. 43-68. doi : 10.5802/tsg.363. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.363/

[1] Ian Agol The virtual Haken conjecture, Doc. Math., Volume 18 (2013), pp. 1045-1087 (With an appendix by Agol, Daniel Groves, and Jason Manning) | MR | Zbl

[2] Hans-Jürgen Bandelt Retracts of hypercubes, J. Graph Theory, Volume 8 (1984) no. 4, pp. 501-510 | DOI | MR | Zbl

[3] Hans-Jürgen Bandelt; Henry M. Mulder; Elke Wilkeit Quasi-median graphs and algebras, J. Graph Theory, Volume 18 (1994) no. 7, pp. 681-703 | DOI | MR | Zbl

[4] Martin R. Bridson; André Haefliger Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999 | DOI | MR | Zbl

[5] Robert G. Burns; Abraham Karrass; Donald M. Solitar A note on groups with separable finitely generated subgroups, Bull. Aust. Math. Soc., Volume 36 (1987) no. 1, pp. 153-160 | DOI | MR | Zbl

[6] Pierre-Emmanuel Caprace; Michah Sageev Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal., Volume 21 (2011) no. 4, pp. 851-891 | DOI | MR | Zbl

[7] Jérémie Chalopin; Victor Chepoi; Hiroshi Hirai; Damian Osajda Weakly modular graphs and nonpositive curvature (2014) (https://arxiv.org/abs/1409.3892)

[8] Victor D. Chepoi Graphs of some CAT (0) complexes, Adv. Appl. Math., Volume 24 (2000) no. 2, pp. 125-179 | DOI | MR | Zbl

[9] Yves Cornulier Group actions with commensurated subsets, wallings and cubings (2013) (https://arxiv.org/abs/1302.5982)

[10] Daniel S. Farley Actions of picture groups on CAT(0) cubical complexes, Geom. Dedicata, Volume 110 (2005) no. 1, pp. 221-242 | DOI | MR | Zbl

[11] Anthony Genevois Cubical-like geometry of quasi-median graphs and applications to geometric group theory, Ph. D. Thesis, Aix Marseille University, France (2017)

[12] Anthony Genevois Embeddings in Thompson’s groups from quasi-median geometry (2017) (https://arxiv.org/abs/1709.03888) | Zbl

[13] Anthony Genevois Lampligther groups, median spaces, and A-T-menability (2017) (https://arxiv.org/abs/1705.00834)

[14] Anthony Genevois Negative curvature of automorphism groups of graph products with applications to right-angled Artin groups (2018) (https://arxiv.org/abs/1807.00622)

[15] Anthony Genevois; Alexandre Martin Automorphisms of graph products of groups from a geometric perspective (2018) (https://arxiv.org/abs/1809.08091) | Zbl

[16] Elisabeth R. Green Graph products of groups, Ph. D. Thesis, University of Leeds, United Kingdom (1990)

[17] Mikhail L. Gromov Hyperbolic Groups, Essays in Group Theory (S. M. Gersten, ed.) (Mathematical Sciences Research Institute Publications), Springer, 1987, pp. 75-263 | DOI | Zbl

[18] Victor S. Guba; Mark V. Sapir Diagram groups, Memoirs of the American Mathematical Society, 130, American Mathematical Society, 1997 no. 620 | DOI | MR | Zbl

[19] Victor S. Guba; Mark V. Sapir On subgroups of the R. Thompson group F and other diagram groups, Sb. Math., Volume 190 (1999) no. 8, pp. 3-60 | DOI | MR | Zbl

[20] Victor S. Guba; Mark V. Sapir Diagram groups are totally orderable, J. Pure Appl. Algebra, Volume 205 (2006) no. 1, pp. 48-73 | DOI | MR | Zbl

[21] Sang-Hyun Kim; Thomas Koberda Embedability between right-angled Artin groups, Geom. Topol., Volume 17 (2013) no. 1, pp. 493-530 | MR | Zbl

[22] Henry M. Mulder The interval function of a graph, Mathematical Centre Tracts, 132, Mathematisch Centrum, Amsterdam, 1980 | MR | Zbl

[23] Ladislav Nebeský Median graphs, Commentat. Math. Univ. Carol., Volume 12 (1971) no. 2, pp. 317-325 | MR | Zbl

[24] Graham A. Niblo; Lawrence D. Reeves The geometry of cube complexes and the complexity of their fundamental groups, Topology, Volume 37 (1998) no. 3, pp. 621-633 | DOI | MR | Zbl

[25] M. Roller Pocsets, median algebras and group actions: An extended study of Dunwoody’s construction and Sageev’s theorem, 1998 (dissertation)

[26] Michah Sageev Ends of group pairs and non-positively curved cube complexes, Proc. Lond. Math. Soc., Volume 71 (1995) no. 3, pp. 585-617 | DOI | MR | Zbl

[27] Michah Sageev; Daniel T. Wise The Tits alternative for CAT(0) cubical complexes, Bull. Lond. Math. Soc., Volume 37 (2005) no. 5, pp. 706-720 | DOI | MR | Zbl

[28] Elke Wilkeit The retracts of Hamming graphs, Discrete Math., Volume 102 (1992) no. 2, pp. 197-218 | DOI | MR | Zbl

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